Approximate Proximity Drawings

  • William Evans
  • Emden R. Gansner
  • Michael Kaufmann
  • Giuseppe Liotta
  • Henk Meijer
  • Andreas Spillner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)


We introduce and study a generalization of the well-known region of influence proximity drawings, called 1,ε 2)-proximity drawings. Intuitively, given a definition of proximity and two real numbers ε 1 ≥ 0 and ε 2 ≥ 0, an (ε 1,ε 2)-proximity drawing of a graph is a planar straight-line drawing Γ such that: (i) for every pair of adjacent vertices u,v, their proximity region “shrunk” by the multiplicative factor \(\frac{1}{1+\varepsilon_1}\) does not contain any vertices of Γ; (ii) for every pair of non-adjacent vertices u,v, their proximity region “blown-up” by the factor (1 + ε 2) contains some vertices of Γ other than u and v. We show that by using this generalization, we can significantly enlarge the family of the representable planar graphs for relevant definitions of proximity drawings, including Gabriel drawings, Delaunay drawings, and β-drawings, even for arbitrarily small values of ε 1 and ε 2. We also study the extremal case of (0,ε 2)-proximity drawings, which generalizes the well-known weak proximity drawing model.


Planar Graph Delaunay Triangulation Adjacent Vertex Outerplanar Graph Planar Embedding 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Angelini, P., Bruckdorfer, T., Chiesa, M., Frati, F., Kaufmann, M., Squarcella, C.: On the Area Requirements of Euclidean Minimum Spanning Trees. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 25–36. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Aronov, B., Dulieu, M., Hurtado, F.: Witness (Delaunay) graphs. Comput. Geom. 44(6-7), 329–344 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aronov, B., Dulieu, M., Hurtado, F.: Witness Rectangle Graphs. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 73–85. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Bose, P., Lenhart, W., Liotta, G.: Characterizing proximity trees. Algorithmica 16(1), 83–110 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Di Battista, G., Lenhart, W., Liotta, G.: Proximity Drawability: a Survey. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 328–339. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  7. 7.
    Di Battista, G., Liotta, G., Whitesides, S.: The strength of weak proximity. J. Discrete Algorithms 4(3), 384–400 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dillencourt, M.B.: Realizability of Delaunay triangulations. Inf. Process. Lett. 33(6), 283–287 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dillencourt, M.B., Smith, W.D.: A Simple Method for Resolving Degeneracies in Delaunay Triangulations. In: Lingas, A., Carlsson, S., Karlsson, R. (eds.) ICALP 1993. LNCS, vol. 700, pp. 177–188. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  10. 10.
    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H.: Drawing a Tree as a Minimum Spanning Tree Approximation. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part II. LNCS, vol. 6507, pp. 61–72. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  11. 11.
    Jaromczyk, J.W., Toussaint, G.T.: Relative neighborhood graphs and their relatives. Proc. IEEE 80(9), 1502–1517 (1992)CrossRefGoogle Scholar
  12. 12.
    Lenhart, W., Liotta, G.: Proximity Drawings of Outerplanar Graphs. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 286–302. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  13. 13.
    Li, X.: Applications of computational geometry in wireless networks. In: Cheng, X., Huang, X., Du, D.-Z. (eds.) Ad Hoc Wireless Networking, pp. 197–264. Kluwer Academic Publishers (2004)Google Scholar
  14. 14.
    Liotta, G.: Proximity drawings. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization. CRC Press (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • William Evans
    • 1
  • Emden R. Gansner
    • 2
  • Michael Kaufmann
    • 3
  • Giuseppe Liotta
    • 4
  • Henk Meijer
    • 5
  • Andreas Spillner
    • 6
  1. 1.University of British ColumbiaCanada
  2. 2.AT&T Research LabsUS
  3. 3.Universität TübingenGermany
  4. 4.Università degli Studi di PerugiaItaly
  5. 5.Roosevelt AcademyThe Netherlands
  6. 6.Universität GreifswaldGermany

Personalised recommendations